Acclato Physics Synchoton Radiation A. Bogacz, G. A. Kafft, and T. Zolkin Jffson Lab Coloado Stat Univsity Lctu 8 USPAS Acclato Physics Jun 13
Synchoton Radiation Acclatd paticls mit lctomagntic adiation. Emission fom vy high ngy paticls has uniqu poptis fo a adiation souc. As such adiation was fist obsvd at on of th alist lcton synchotons, adiation fom high ngy paticls (mainly lctons is known gnically as synchoton adiation by th acclato and HENP communitis. Th adiation is highly collimatd in th bam diction Fom lativity ct ' γ ct γβ z x' x y' y z' γβct + γ z USPAS Acclato Physics Jun 13
Lontz invaianc of wav phas implis k µ (ω/c,k x,k y,k z is a Lontz 4-vcto ω γω γβ kc z k x kx k k y y k γβω / c+ γ k z kx + ky k x + k y k z sin θ sin θ cosθ ω / c ω / c ω / c ω / c γω / c+ γβk γ 1+ β cos θ ω / c z z ( ( ( ( ω / c γω / c γβk γ 1 β cos θ ω / c z USPAS Acclato Physics Jun 13
θ sinθ sinθ γ β θ ( 1+ cos Thfo all adiation with θ' < π /, which is oughly ½ of th photon mission fo dipol mission fom a tansvs acclation in th bam fam, is Lontz tansfomd into an angl lss than 1/γ. Bcaus of th stong Doppl shift of th photon ngy, high fo θ, most of th ngy in th photons is within a con of angula xtnt 1/γ aound th bam diction. USPAS Acclato Physics Jun 13
Lamo s Fomula Fo a paticl xcuting non-lativistic motion, th total pow mittd in lctomagntic adiation is (Lamo, vifid lat 1 q 1 P ( t a p & 6πε c 6πε m c 3 3 Linad s lativistic gnalization: Not both de and dt a th fouth componnt of lativistic 4-vctos whn on is daling with photon mission. Thfo, thi atio must b an Lontz invaiant. Th invaiant that ducs to Lamo s fomula in th non-lativistic limit is µ du duµ P 6πε c dτ dτ USPAS Acclato Physics Jun 13
( P t 6 c γ & πε β β & β 6 Fo acclation along a lin, scond tm is zo and fist tm fo th adiation action is small compad to th acclation as long as gadint lss than 1 14 MV/m. Tchnically impossibl. & β c Fo tansvs bnd acclation β ˆ ρ ( P t c 6πε ρ β γ 4 4 USPAS Acclato Physics Jun 13
Factional Engy Loss δ E Θβ γ 6πε ρ 3 4 Fo on tun with isomagntic bnding filds δ E E bam 4π 3ρ β γ 3 3 is th classical lcton adius:.8 1-13 cm USPAS Acclato Physics Jun 13
Radiation Pow Distibution Consulting you favoit Classical E&M txt (Jackson, Schwing, Landau and Lifshitz Classical Thoy of Filds dp d 3 ω γ 8 ω π ε ρ ωc ω / ω c K 5/3 ( xdx USPAS Acclato Physics Jun 13
Citical Fquncy Citical (angula fquncy is 3 3 c ωc γ ρ Engy scaling of citical fquncy is undstood fom 1/γ mission con and fact that 1 β ~1/( γ t A ρ γβ c A B ρ ρ ρ t t B γ γ γ 3 3 c c c t B ρ ρ ρ + γβ γ γ 3 c c c 1/γ USPAS Acclato Physics Jun 13
Photon Numb dp 3 c c 5/3 ( dω 8π ε ρ 6πε ξ ρ P dω ωγ ξ K x dxdξ γ dn& 1 dp dω hω dω 4 hω dn& hω d ω dω dn& d ω dω 8 15 3 hω c n& 5α c 5α 1 γ δn γ α 3 ρ 3 Θ 4πε hc 137 USPAS Acclato Physics Jun 13
Instion Dvics (ID Oftn piodic magntic fild magnts a placd in bam path of high ngy stoag ings. Th adiation gnatd by lctons passing though such instion dvics has uniqu poptis. Fild of th instion dvic magnt B x y z B z y B z B z ( ( ˆ ( ( π λ,, cos / ID Vcto potntial fo magnt (1 dimnsional appoximation B λ A x y z A z x A z z π ( ( ( ID,, ˆ sin ( π / λ ID USPAS Acclato Physics Jun 13
Elcton Obit Unifomity in x-diction mans that canonical momntum in th x-diction is consvd ( A z K vx ( z csin z/ γm γ ( π λ ID v 1 K λ x z dz z v β γ π x ID ( cos( π / λ Fild Stngth Paamt z z ID K B λ ID π mc USPAS Acclato Physics Jun 13
Avag Vlocity Engy consvation givs that γ is a constant of th motion 1 βz β x βz βx γ ( z 1 ( z ( z Avag longitudinal vlocity in th instion dvic is β Avag st fam has β z 1 1 γ K γ γ 1 1 β γ 1+ K / USPAS Acclato Physics Jun 13
Rlativistic Kinmatics In avag st fam th instion dvic is Lontz contactd, and so its wavlngth is λ λ ID / β γ Th sinusoidal wiggling motion mits with angula fquncy ω πc / λ Lontz tansfomation fomulas fo th wav vcto of th mittd adiation k k k k x y z γ k k k x y γ k ( 1 β cosθ k sinθ cosϕ k sinθ sinϕ ( cosθ β USPAS Acclato Physics Jun 13
ID (o FEL Rsonanc Condition Angl tansfoms as ( cosθ β ( 1 β cosθ k cos θ z k Wav vcto in lab fam has k γ k πβ c ( 1 β cosθ λ ( 1 β cosθ ID In th fowad diction cos θ 1 λid λid λ 1 / γ γ + ( K USPAS Acclato Physics Jun 13
Pow Emittd Lab Fam Lamo/Linad calculation in th lab fam yilds P 4 K π γ βzc πε γ λid 1 6 Total ngy adiatd aft on passag of th instion dvic δe π γ β z NK 6πε λid β USPAS Acclato Physics Jun 13
Pow Emittd Bam Fam Lamo/Linad calculation in th bam fam yilds P π 1 ck 6πε λ Total ngy of ach photon is ħπc/λ, thfo th total numb of photons adiatd aft on passag of th instion dvic N π γ αnk 3 USPAS Acclato Physics Jun 13
Spctal Distibution in Bam Fam Bgin with pow distibution in bam fam: dipol adiation pattn (singl hamonic only whn K<<1; plac γ by γ, β by β dp dω c Kk sin Θ 3πε Numb distibution in tms of wav numb Solid angl tansfomation dnγ α k + k dω 4 k d y z NK Ω dω ( γ 1 βcosθ USPAS Acclato Physics Jun 13
Numb distibution in bam fam dn dω Engy is simply E ( θ + ( 4 ( γ α sin θ sin ϕ γ cosθ β NK 4 4 γ 1 βcosθ πβc h Eˆ ( θ 1 λ 1 βcosθ 1 βcosθ ID ( ( Numb distibution as a function of nomalizd lab-fam ngy dn ˆ γ απ E NK 1 β ˆ 3 + de 4γ β γ USPAS Acclato Physics Jun 13
Limits of intgation Avag Engy ˆ 1 ˆ 1 cosθ 1 E cosθ 1 E 1 β 1+ β Avag ngy is also analytically calculabl E dnγ ˆ E de deˆ γ h πβc/ λ dn de ˆ deˆ max ID γ E USPAS Acclato Physics Jun 13
Numb Spctum θ π θ sin θ 1/ γ dn/de γ E und h π c/ λ ID βγ E und (1+ββγ E und E γ USPAS Acclato Physics Jun 13
Convntions on Foui Tansfoms Fo th tim dimnsions i t f% ω f t dt ( ω ( 1 iω t f ( t f% ( ω dω π Rsults on Diac dlta functions ( ( % δ ω δ δ 1 iω t t dω π ( iω t t dt 1 USPAS Acclato Physics Jun 13
Fo th th spatial dimnsions δ ik x 3 f% k f x d x ( ( f ( x 1 ( + ik x 3 f% k d k 3 ( π x x d k 3 1 + ik x 3 ( δ ( 3 ( π USPAS Acclato Physics Jun 13
Gn Function fo Wav Equation Solution to inhomognous wav quation 1 + + x y z c t G( x, t; x, t 4πδ ( x x δ ( t t Will pick out th solution with causal bounday conditions, i.. G x, t; x, t t < t ( This choic lads automatically to th so-calld Rtadd Gn Function USPAS Acclato Physics Jun 13
In gnal G( x, t; x, t t < t G( x, t; x, t ( i( k x ωt ( i( k x+ ωt 3 A k + B k d k t > t bcaus th a two possibl signs of th fquncy fo ach valu of th wav vcto. To solv th homognous wav quation it is ncssay that ω k k c ( i.., th is no dispsion in f spac. USPAS Acclato Physics Jun 13
Continuity of G implis ω A k B k Intgat th inhomognous quation btwn and t t ε 1 G( x, t; x, t 4πδ ( x x c t ( ( i t iω t ( π t + ε ( i( k x ωt ( i( k x+ ωt 3 iω A k + iωb k d k 4πc δ x x ( c A k iω ik x iω t t t + ε ( USPAS Acclato Physics Jun 13
G ( x, t; x, t c ( π i ik x x c iω ( t t c dk π x x π δ ( x x / c t + t + x x 1 i( k ( x x ω( t t i( k ( x x + ω( t t 3 d k ω t > t x ik x x x ( > + iω t t dk t t Calld tadd bcaus th influnc at tim t is du to th souc valuatd at th tadd tim t t x x / c USPAS Acclato Physics Jun 13
Rtadd Solutions fo Filds 1 ρ + + x y z c t φ ε 1 + + A µ J x y z c t 1 3 ρ ( x, t φ ( x, t d x dt δ x x / c t + t 4πε x x µ 3 J ( x, t A ( x, t d x dt δ ( x x / c t + t 4π x x ( Tip: Lav th dlta function in it s intgal fom to do divations. Don t hav to mmb complicatd dlta-function uls USPAS Acclato Physics Jun 13
φ Dlta Function Rpsntation (, 3 x t d x dt dω 8π ε x x (, 1 µ 3 A x t d x dt dω 8π ρ J ( x, t iω x x / c ( t t ( x, t iω x x / c ( t t Evaluation can b xpditd by noting and using th symmty of th Gn function and using lations such as x x x f t t f t t t t f x x f x x x ( ( ( ( USPAS Acclato Physics Jun 13
Radiation Fom Rlativistic Elctons φ (, 1 3 x t d x dt dω 8π ε x x ( ( ( x, t iω x x / c ( t t µ 3 J ( x, t iω x x / c ( t t A( x, t d x dt dω 8π x x ρ δ δ ( ( ( ( ( 3 3 x, t q x t J x, t qv t x t q 1 φ( x, t dt dω 8π ε x qµ v A( x, t dt dω 8π x ρ ( t ( t ( t iω x t c t t ( / ( iω x t c t t ( / ( USPAS Acclato Physics Jun 13
Linad-Wicht Potntials φ ( x, t A x t φ (, ( x, t A x t (, q 4πε qµ 4π δ dt dt ( / ( x ( t ( x t c t t v / ( t δ x ( t c ( t t x ( t q 1 4πε x t nˆ t qµ 4π x t nˆ t ( ( 1 β ( ( v ( t ( 1 β ( ( t t USPAS Acclato Physics Jun 13
EM Fild Radiatd q 1 iω x ( t / c ( t t φ( x, t dt dω 8π ε x ( t qµ v ( t iω x ( t / c ( t t A( x, t dt dω 8π x ( t A E φ B A t & ˆ nˆ q n β q E + ( 3 4πε γ 1 nˆ β R 4πε c ( 1 nˆ 3 β R t B nˆ E / c Vlocity Fild Acclation Fild {( nˆ β β} t USPAS Acclato Physics Jun 13
1 nˆ x ( t nˆ x ( t x ( t d 1 nˆ β c dnˆ d / dt + nˆ n d / dt dt x t dt x t ( x ( t ( q nˆ 1 iω x ( t / c φ ( x, t dt dω 8π ε x ( t qµ v iω x ( t / c ( t t A( x, t dt dω t 8π x ( t iω ( t d ω ( ( i ω 1 β t n ˆ t dt ( ( ( ( ˆ ( i x t / c t t iω x ( t / c ( t t iω x t c t t L ( / ( USPAS Acclato Physics Jun 13
q iω ( / ( ˆ x t c t t n E( x, t dt dω + 8π ε x ( t intgat by pats to gt final sult i x t c t t ( / ( ω q E( x, t dt dω vl 8π ε ˆ ( 1 β n x ( t ( 1 β nˆ + β nˆ + β nˆ β nˆ + β nˆ β nˆ nˆ ( β + β nˆ ( ( ( β 1 β nˆ + β nˆ β nˆ β + β nˆ ( ( iω cx t ( β nˆ ( USPAS Acclato Physics Jun 13
( ˆ & n β nˆ & β ( 1 β nˆ β ( & β nˆ ( / ( ( 1 β ( ( / ( iω x t c t t q E( x, t dt dω acc 8π ε c nˆ x t iω x t c t t q dt dω nˆ & n β β 8π ε c nˆ x t ( 1 β ( {( ˆ } USPAS Acclato Physics Jun 13
Lamo s Fomula Vifid Fo small vlocitis can nglct tadation q E( x, t ˆ { n nˆ & β } / R acc 4πε c dp dω P q nˆ 16π ε µ c 3 v sin Θ 3 16π ε c q 6πε c q 3 v & & & { nˆ β } Θ Angl btwn acclation and popagation dictions Classical Dipol Radiation Pattn USPAS Acclato Physics Jun 13
max ( ( Rlativistic Paking In fa fild aft shot acclation ( ( ˆ ˆ & dp t n n β q 5 dω 16π ε c nˆ dp t θ dω 1 γ { β} ( 1 β q & β sin θ π ε β θ ( 16 c 1 cos Fo cicula motions dp t q & β sin θ cos ϕ 1 dω 16π ε c 1 β cosθ γ 1 β cosθ 5 ( ( 3 USPAS Acclato Physics Jun 13
Spctum Radiatd by Motion de dp 1 ( dt E H nr ˆ dt E E R dt dω dω cµ {( } {( } ˆ ˆ & ˆ ˆ 1 q n n β β & n n β β ( t ( t ( ( cµ 8π ε c 1 nˆ β 1 nˆ β ( ( ( iω R 1 nˆ t / R nˆ t / R / c t t iω + + R 1 nˆ ( t / R+ ( nˆ ( t / R / c t+ t claing th unpimd tim intgal and omga pim dt dωdt dω dt intgal with dlta psntation {( } ˆ ˆ & ˆ ˆ π q n n β β n n β β ( t ( cµ 8π ε c 1 nˆ β ( 1 nˆ β iω nˆ ( t / c+ t iω nˆ ( t / c+ t dt dt dω {( &} ( t USPAS Acclato Physics Jun 13
ˆ d E q n 3 dωdω 3π ε c nˆ β {( nˆ β β} ( 1 & iω nˆ ( t / c t+ t dt d E ω Ω q ω ( iω t nˆ ( t / c nˆ nˆ β dt 3π ε 3 d d c Facto of two diffnc fom Jackson bcaus h combins positiv fquncy and ngativ fquncy contibutions in on positiv fquncy intgal. I don't lik bcaus Pasval's fomula, tc. don't wok! W'll pfom this calculation in nw intnsity gims of pulsd lass. USPAS Acclato Physics Jun 13